表 面 物 理 学

1. 表 面 物 理 学 江颖 量子材料中心

2. 表面热力学和动力学 第六课:

3. 本课内容提要： (一)：表面热力学 表面热力学基本方程 表面张力和表面应力 晶体的表面能和平衡形状 表面的生长模式 (二): 表面动力学 表面生长动力学 表面二维岛的生长 表面二维和三维岛的退化

4. 将化学反应应用于生产实践主要有两个方面的问题：将化学反应应用于生产实践主要有两个方面的问题： 了解反应进行的方向和最大限度以及外界条件对平衡的影响。 知道反应进行的速率和反映的历程（即机理）。 人们把前者归属于热力学的研究范围，把后者归属于动力学的研究范围。 化学中的热力学和动力学

5. 第一部分： 表面热力学 Josiah Willard Gibbs (1839-1903)

6. 前面主要讨论了“静止”表面的结构。这一节将从宏观角度，运用平衡热力学基本理论研究晶体表面的各种热现象。前面主要讨论了“静止”表面的结构。这一节将从宏观角度，运用平衡热力学基本理论研究晶体表面的各种热现象。 研究宏观热现象的基本方法，是在实验的基础上，用能量的观点研究温度对表面结构和表面性质的影响。以平衡态的统计热力学理论为基础，研究表面热现象及各种特征参数之间的关系，探讨有关表面热力学、表面物理化学过程的基本规律。

7. Kinetic Processes and Surface in Equilibrium Surface is no static system. The characteristic bonding energy of an atom at a special site and the energy barriers between such sites determine the probability for adsorption, desorption and diffusion of atoms at a given temperature. Fig. 1. Schematic representation of fundamental atomic processes occuring during epitaxial growth.

8. Principle of detailed balance In the thermodynamic equilibrium all surface processes proceed in two opposite directions at equal rates according to the “principle of detailed balance”. Detailed balance means that the rate constants for the forward direction, rf, and the backward direction, rb, of a process satisfy the relation： rf /rb = exp(-ΔE/kBT), where ΔE is the energy difference between the initial and final states. Processes, such as adsorption and desorption, decay and formation of islands, etc. must obey the detailed balance. A surface after stopping growth for while at not too high temperature can be considered to be in thermodynamic equilibrium with the substrate and the surroundings, e.g., the rest gas. Crystal growth is clearly related to non-equilibrium kinetic processes. Yet the “principle of detailed balance” is still fulfilled.

9. dU =(U/ S)V,N dS + ( U/V)S,N dV + ( U/N)S,V dN dU = T dS – P dV + μ dN The extensive property of the internal energy: U(λ S, λ V, λ N) = λ U(S,V,N) U = T S – P V + μN S dT – V dP + N d μ = 0 (Gibbs-Duhem 等式) Internal energy(内能): U = U(S,V,N) 常用热力学函数: Enthalpy(焓): H = U + PV = T S + μN Helmholtz free energy (自由能): F = U - TS = - P V + μN Gibbs 函数(Gibbs 自由能) ： G = F + PV =μN Grand potential (巨势，自由能的差)Ω= F - G = - P V 1

10. For equilibrium system: T, P, μ are identical in different phases, S = S1 + S2 + Ss V = V1 + V2 + Vs N = N1 + N2 + Ns 表面的热力学基本方程 U = T S – P V + μN U = T S – P V + μN + γA γ = (U/ A)S,P,N -surface tension (表面能也称作表面张力) Enthalpy(焓): H = U + PV = T S + μN + γA Helmholtz free energy (自由能): F = U - TS = - P V + μN+ γA Gibbs 函数(Gibbs 自由能) ： G = F + PV - γA =μN Grand potential(巨势，自由能的差):Ω= F - G = - P V+ γA 2

11. 吉布斯定义的表面位置: V = V1 + V2 N = ρ1V1 + ρ2V2 In comparison to the total number of particles we have Ns = 0 & Vs=0 Enthalpy(焓): H = U + PV = T S + μN + γA Helmholtz free energy (自由能): F = U - TS = - P V + μN+ γA Gibbs 函数(Gibbs 自由能) ： G = F + PV - γA =μN Grand potential(巨势，自由能的差):Ω= F - G = - P V+ γA Gs = 0 Ω = Ω1 + Ω2 + Ωs= - P (V1+V2)+ γ A Ωs= γ A

12. 表面张力 = 表面应力？ Surface tension Surface stress 液体：Yes！ 固体：No！

13. U = T S – P V + μN + γA γ = (U/ A)S,V,N σ dU = A Σijσij dεij Linear elasticity theory ε Surface strain (应变张量) Surface stress (应力张量) dU = T dS + S dT– P dV – V dP + μ dN + N dμ + γ dA + A d γ dU = T dS – P dV + μ dN + A Σijσij dεij S dT– V dP + N d μ + γ dA + A d γ - A Σijσij dεij = 0 dA /A = Σ d εijδij A d γ + S dT– V dP + N d μ + A Σ ( γij δij- σij )d εij = 0 A d γ + (S1+S2 +Ss) dT– (V1 + V2 + Vs)dP + (N1 + N2 + Ns) d μ + A Σ ( γij δij - σij )d εij = 0 (Gibbs-Duhem 等式) Si dT – Vi dP + Ni dμ = 0; i=1,2 A d γ + Ss dT– Vs dP + Ns d μ + A Σ ( γij δij - σij )d εij = 0 - Gibbs　吸附方程

14. A d γ + Ss dT– Vs dP + Ns d μ + A Σ ( γij δij - σij )d εij = 0 - Gibbs　吸附方程 V = V1 + V2 N = ρ1V1 + ρ2V2 In comparison to the total number of particles we have Ns = 0 & Vs=0

15. Ss = -A(γ/T)ε γ = γ(ε,T) σij = γδij+ ( γ/ εij)T A special example is liquid surface where the surface atoms is free to rearrange themselves: σ = γ 若：  γ/ ε< 0

16. Equilibrium Shape of Small Crystals Anisotropy of Surface Energy The surface free energy per unit area, γ, of a certain crystal surface varies with its crystallographic orientation characterized by the surface plane (hkl) or the surface normal n, i.e., γ=γ (hkl) or γ=γ(n). 晶体达到平衡时，其表面能为各晶面表面能的总和： ∑ γ (hkl)dAhkl 若以θ表示晶面的方向角，则表面能γ随方向角θ的不同而改变。根据表面能的方向性推测晶体的平衡形状，最成功的方法是Wulff作图法。经过数学上的严格证明，这一方法得到了公认。 3

17. Fig. 3 shows the orientation dependence. Its nominal (1n) surface (n>>1) represents a vicinal (01) surface. It consists of a high number of (01) terraces separated by atomic steps of height a. With θ~1/n as the angle of orientation of [1n] against the [01] orientation, the step density is given as tan θ/a. If βs is the energy per step and γ(0) is the energy of a (01) face, the surface energy of a (1n) surface is γ(θ) = cosθγ(0)+βs(sinθ/a) The prefactor cosθ guarantees that the relative amount of the (01) terraces to the total surface area reduces with increasing angle θ. The interaction of steps has been neglected here. The increase of the angle from θ=0 to large values is accompanied by an increase of the step density. A proper expression for γ(θ) must hence include the interaction between steps. Fig. 3 A (1n) surface, which is slightly misoriented from the (01) surface.

19. Wulff Construction Wulff Theorem: The equilibrium crystal shape (ECS) at constant temperature T with fixed volume V and chemical potential μ is determined by the minimal excess surface free energy with respect to the surface A Fs = A(V)γ(n)dA Subject to the constraint of fixed volume V = V(A)dV. The theorem states that the ECS is not necessarily that of the minimum surface area. It may be a complex polyhedron with the lowest total surface energy for a given volume. In the case of crystals the variation of γ with the normal n will produce, on each surface element dA, a force proportional to ∂γ/∂n, which will tend to alter its direction at the same time as γ tends to shrink its area. A minimal surface only occurs for a perfectly spherical Wulff plot, i.e., an isotropic excess surface free energy. The corresponding ECS is a sphere.

20. Wulff作图法 表面能是一标量，为了表示其与晶面方向的相关性，引入一矢量 ,矢量的方向为晶面的法线方向，其长度正比于表面能数值。因此，(hkl)即为表面能γ (hkl)的矢径。

21. Fig.9 A polar plot of the surface free energy for a 2D crystal (solid line) and the ECS based on the Wulff construction (dotted line). The higher the γ(n), the smaller the corresponding surface. In theoretical (table 2.1), γ(111)<γ(311)<γ(100)<γ(110), so (110) surface is negligible. Fig.10 Equilibrium shapes of a Si crystal based on the Wulff construction using (a) exp. values or (b) theo. values. (100), (311), (110), and (111), from black to white, surface orientations are considered. (Surface energies from Table 2.1).

22. Surface Energy and Morphology Facetting and Roughening The surface buckling happens on a mesoscopic length scale (large than atomic distances). For a small buckling surface, we have F’s = A’γ(θ)dA’ = Aγ(θ)dA/cos (θ) The assumption of a weak variation of γ withθyields an expansion of the integrand up to second order, F’s = γ(0)A +Adγ/d θ | θ=0+ ½ A θ2[γ(0)+ d2γ/d θ 2 | θ=0 ]dA The first term gives the energy of the flat surface. The second one vanishes for symmetry reasons. The third term gives the energy due to surface buckling. For γ(0) +d2γ/d θ 2 | θ=0 >0, the flat surface is stable (or at least metastable), for its <0, the buckling surface is more stable. Fig.11 Small buckling of a surface.

23. 表面上薄膜生长的三种典型模式 SK 生长 岛状生长 层状生长

24. γs -the surface free energy of substrate-vacuum interface γo - the overlayer-vacuum interface γs/o - is the substrate-overlayer interface The equilibrium of the forces holds as γs = γs/o + γo cosφ with φ as the angle between the overlayer-vacuum face and the substrate-vacuum face. Set Δγ=γs/o + γo – γs For Frank-van der Merve (2D) mode, we have Δγ 0 with φ = 0; For Volmer-Weber (3D) mode, we have Δγ> 0 with φ > 0; For Stranski-Krastanov mode, we have Δγ 0 the first atomic layers (wetting layer) and Δγ> 0 for the islands.

25. 第二部分：表面动力学

26. 一般来说，上述表面热力学理论只能描述处于平衡状态下的表面形貌，而在实际的薄膜生长过程中，体系往往是处于非平衡状态，这时候则需要用非平衡动力学的观点来描述。

27. 3D Versus 2D Growth In thermodynamic equilibrium, there is no net growth. The crystal growth must be a non-equilibrium kinetic process. The resulting macroscopic state of the system depends on the reaction paths in the configuration space. Since the result is kinetically determined, the obtained state is not necessarily the most stable one.

28. The thermal stability of the Al thin films Initial surface 24 hours later( 500nm x 500nm) 2.4ML (300nm x 300nm) Thermally annealed (RT) Al film develops various heights enabling the comparison of the relative stability of islands with different heights

29. Kinetics is a concept only involved with the movement of objects. Dynamics focuses on the forces and their effects. In certain point of view, we can say kinetics only deals with the motions of objects, and dynamics with the reason why and how they moves. 动力学

30. We can consider the growth on the surface as a kind of chemical reaction. Kinetics: the influence of external macroscopic variables on the overall reaction rate (temperature, pressure, relative concentration) Dynamics: the detailed atomic motions that characterize an elementary act of reaction. Sometimes these two terms are not distinguishable. 表面动力学

31. Generic Understanding of Growth Growth, by definition, is nonequilibrium in nature, and in many cases is far from equilibrium. A specific growth mode is selected by the interplay between thermodynamics and growth kinetics. Thermodynamics Kinetics (various atomic rate processes)

32. Atom Dynamics & Kinetics A: formation of a surface vacancy-adatom pair,or their recombination. B: association or dissociation of adatoms with an atomic cluster & cluster diffusion. C: diffusion of a surface vacancy ,especially toward the lattice step. D: falling off a lattice step of an adatom. E: diffusion of an adatom & its long range interactions with other adatoms. G: diffusion,dissociation & activation of a ledge atom. H: dissociation & activation of a kink atom.

33. 实验表明，参与表面各种原子过程的原子扩散能力可以用表面扩散系数来描述。表面扩散系数与扩散原子的跳跃几率有关,可以表示为:实验表明，参与表面各种原子过程的原子扩散能力可以用表面扩散系数来描述。表面扩散系数与扩散原子的跳跃几率有关,可以表示为: 这里D0是尝试频率 (1012-1013), Vs是能量势垒, kB是玻耳兹曼常数, T 是温度。

34. 表面原子的扩散 -长程跳步 (jump over long distance) 原子每次跳步的距离为晶格常数的整数倍， 即原子每次跳步的长度大于一个最近邻位。 -交换扩散机制 跳步扩散 (hopping)或替换扩散(exchange)。替换扩散机制产生的物理原因是由于系统应保持沿扩散路径的断键数目最少。 初态 过渡态 末态 跳步扩散 替换扩散

35. Philosophy If we can establish EVERY correspondence between Atomic Rate Process Morphological Evolution then in principle we should be able to select a preferred growth mode via precise control of the various rate processes.

36. Two growth models - Step flow growth Burton et al, Phil. Trans. R. Soc. London Ser. A 243 299(1951) 台阶指的是两层台面(terrace)之间的边界，沿着晶体某一指数面切割表面通常都会在表面产生台阶，这种含有台阶的面叫做邻面(vicinal surface)。由于表面上原子的吸附能力强烈地依赖于吸附位的最近邻原子数，而位于台阶处的原子比位于表面上的原子具有更多的配位数，因此沉积原子与台阶的键合更强。这一理论假定在邻面上，如果沉积原子运动速率较高，在新的沉积原子到来之前基底上已有的扩散单原子就能够到达台阶处并与之结合。这样一来，通过沉积原子和台阶的键合使台阶不断前进，从而使生长连续进行，这种生长叫做模式叫做step flow模式 。较高温度和较低的沉积速率时更容易造成形成这种生长模式。

38. (沉积原子运动快)

40. Two growth models - 形核和成岛生长模式 一般薄膜的生长温度较低，这样沉积原子在基底上的运动较慢，在新的沉积原子到来之前基底上的原子不能运动到台阶处。这些沉积原子就会在表面（台面）上游走，在原子行走过程中，如果能碰到同类原子，它们便结合在一起，形成原子团。如果这个原子团满足一定的能量关系，就会增加它们在基底表面上的居留时间，就有与其它原子集结的更大几率，这就是成核。成核以后形成的原子团并不是稳定的，还存在原子团的离解过程。只有在一定条件下满足一定的能量关系，原子团才不再离解，随外来原子加入或热处理，原子团不断长大。这个一定数量原子构成的原子团即为前述的临界核。达到临界核以后，若继续入射原子到达基底表面，那么原子团不断长大成粒子簇，即岛。这样生长就会通过岛的长大,结合方式进行。

41. 临界尺寸 在经典成核理论中，一个晶核的形成主要取决于Gibbs自由能的变化量。随着晶核尺寸的增加，会出现一个临界晶核尺寸nc，使得Gibbs自由能的变化量取极大值。小于这个尺寸的晶核随着粒子数的增加Gibbs自由能的变化量不断增加，晶核长大的几率比退化的几率小，是不稳定晶核；相反，大于这个尺寸的晶核，随着粒子数的增加Gibbs自由能的变化量不断下降，晶核长大的几率比 退化的几率大，是稳定晶核，如图 所示。在数学上，临界晶核尺寸可 以这样确定:

43. 临界尺寸 在二维亚单层生长中，假定已知临界尺寸为i, 由速率方程可以导出标度关系 这里N是总的岛密度，D是原子在基底表面上的扩散速度，F是沉积速率，Ei是键能，i是临界尺寸。运用方程， 通过实验测量就可以推导出微观参数。比如，测量岛密度随沉积流量的变化规律，可以得到临界尺寸大小i ， 再测量岛密度随温度的变化规律，那么由已知的i , 就可以得到扩散势垒Ed和前因子D0的大小。Monte Carlo模拟和实验上都已经证实了标度关系的存在。

45. 休息15分钟

46. 亚单层生长时表面上典型的二维岛 当沉积流量F=0.167ML/s,覆盖率Θ=0.12ML时, 在Ag/Pt(111)系统中得到的二维岛STM像。 (a)=110K, (b)=280K 。

47. Diffusion-Limited Aggregation ( DLA ) Witten and Sander, Phys. Rev. Lett.47, 1400 (1981)

48. Citation > 3100

49. Nucleation without surfactant Diffusion-Limited-Aggregation (DLA) Hit and Stick 岛的平均分支宽度 b (average brach thinkness) 为1个原子宽度 ( b≈1)。 Witten and Sander, Phys. Rev. Lett.47, 1400 (1981)

50. Nucleation without surfactant Extended Diffusion Limited Aggregation Adatoms can relax along island edges. 这时候沉积原子会沿着岛边缘扩散，稳定位置是近邻原子数大于等于2的位置。 在三角格子上，就会得b为4个原子宽度的分形岛; 在正方格子上不存在这个生长区域。 Zhang, Chen, and Lagally, Phys. Rev. Lett.73, 1829 (1994)